Best Known (55, 55+49, s)-Nets in Base 32
(55, 55+49, 240)-Net over F32 — Constructive and digital
Digital (55, 104, 240)-net over F32, using
- t-expansion [i] based on digital (51, 104, 240)-net over F32, using
- 5 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
- 5 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(55, 55+49, 513)-Net in Base 32 — Constructive
(55, 104, 513)-net in base 32, using
- t-expansion [i] based on (46, 104, 513)-net in base 32, using
- 4 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 4 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(55, 55+49, 1142)-Net over F32 — Digital
Digital (55, 104, 1142)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32104, 1142, F32, 49) (dual of [1142, 1038, 50]-code), using
- 105 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 1, 6 times 0, 1, 14 times 0, 1, 28 times 0, 1, 48 times 0) [i] based on linear OA(3294, 1027, F32, 49) (dual of [1027, 933, 50]-code), using
- construction XX applied to C1 = C([1022,46]), C2 = C([0,47]), C3 = C1 + C2 = C([0,46]), and C∩ = C1 ∩ C2 = C([1022,47]) [i] based on
- linear OA(3292, 1023, F32, 48) (dual of [1023, 931, 49]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,46}, and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3292, 1023, F32, 48) (dual of [1023, 931, 49]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,47], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3294, 1023, F32, 49) (dual of [1023, 929, 50]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,47}, and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(3290, 1023, F32, 47) (dual of [1023, 933, 48]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,46], and designed minimum distance d ≥ |I|+1 = 48 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,46]), C2 = C([0,47]), C3 = C1 + C2 = C([0,46]), and C∩ = C1 ∩ C2 = C([1022,47]) [i] based on
- 105 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 1, 6 times 0, 1, 14 times 0, 1, 28 times 0, 1, 48 times 0) [i] based on linear OA(3294, 1027, F32, 49) (dual of [1027, 933, 50]-code), using
(55, 55+49, 911165)-Net in Base 32 — Upper bound on s
There is no (55, 104, 911166)-net in base 32, because
- 1 times m-reduction [i] would yield (55, 103, 911166)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 107264 379854 343207 687734 933753 350715 556741 574079 403321 921558 473818 992437 451215 883186 581243 383431 651955 654702 005813 867812 310599 162520 861283 560945 251626 108817 > 32103 [i]